can someone help me with the problems below. Thanks

Question

wintersuntime · Answer

attachment

wintersuntime · Answer

@nincompoop

wintersuntime · Answer

@dan815

wintersuntime · Answer

@mathmale

jim_thompson5910 · Answer

for #11, look for `10` in the `f(x)` row. The value above it is the corresponding x value and the answer to that question. In this case, there is going to be more than one answer, so I'd write it as a set of values. In this case, the inverse isn't really a function.

wintersuntime · Answer

okay

mathmale · Answer

Another way to look at this situation involving a function and its inverse:

If    y=f(x), then   $$x=f ^{-1}(y)$$

the table shows that if x=2, y=10, or:$$If.x=2,f(x)=10$$

Now, dealing with the inverse of f(x):$$f ^{-1}(y)=x \rightarrow f ^{-1}(10)=2$$

wintersuntime · Answer

oh alright so they're just switching

mathmale · Answer

However, Jim makes a very important point.  There happen to be two 10's in the 2nd row.  This means $$f ^{-1}(10) $$ is associated not only with 2 in the first row, but also with 6.

mathmale · Answer

So, I would have to conclude that $$f ^{-1}(x)$$

mathmale · Answer

is NOT a function, according to the definition of a function (only one output for any given input).

wintersuntime · Answer

okay

wintersuntime · Answer

so we can't find  $$f ^{-1}(10)$$

wintersuntime · Answer

or would it be equal to 2

mathmale · Answer

Strictly speaking, you can't; I'd use the word "ambiguous."

Had you seen the 2nd '10' first, then you might have replied that the answer was 6.
But we saw the first '10' first and replied that the answer was 2.   Pretty ambiguous!

Strictly speaking     our inverse function   is not a function, because it doesn't satisfy the "only one output for each distinct input" rule.

wintersuntime · Answer

oh okay

mathmale · Answer

Best wishes.  I need to get off the 'Net very soon.   Good night!

wintersuntime · Answer

okay thanks